Homotopy based solutions of the Navier–Stokes equations for a porouschannel with orthogonally moving walls

Hang Xu,1,a� Zhi-Liang Lin,1 Shi-Jun Liao,1,b� Jie-Zhi Wu,2,c� and Joseph Majdalani2,d�1State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering,Shanghai Jiao Tong University, Shanghai 200240, China2University of Tennessee Space Institute, Tullahoma, Tennessee 37388, USA

�Received 5 May 2009; accepted 24 February 2010; published online 4 May 2010�

This paper focuses on the theoretical treatment of the laminar, incompressible, and time-dependentflow of a viscous fluid in a porous channel with orthogonally moving walls. Assuming uniforminjection or suction at the porous walls, two cases are considered for which the opposing wallsundergo either uniform or nonuniform motions. For the first case, we follow Dauenhauer andMajdalani �Phys. Fluids 15, 1485 �2003�� by taking the wall expansion ratio � to be time invariantand then proceed to reduce the Navier–Stokes equations into a fourth order ordinary differentialequation with four boundary conditions. Using the homotopy analysis method �HAM�, an optimizedanalytical procedure is developed that enables us to obtain highly accurate series approximations foreach of the multiple solutions associated with this problem. By exploring wide ranges of the controlparameters, our procedure allows us to identify dual or triple solutions that correspond to thosereported by Zaturska et al. �Fluid Dyn. Res. 4, 151 �1988��. Specifically, two new profiles arecaptured that are complementary to the type I solutions explored by Dauenhauer and Majdalani. Incomparison to the type I motion, the so-called types II and III profiles involve steeper flow turningstreamline curvatures and internal flow recirculation. The second and more general case that weconsider allows the wall expansion ratio to vary with time. Under this assumption, the Navier–Stokes equations are transformed into an exact nonlinear partial differential equation that is solvedanalytically using the HAM procedure. In the process, both algebraic and exponential models areconsidered to describe the evolution of ��t� from an initial �0 to a final state �1. In either case, wefind the time-dependent solutions to decay very rapidly to the extent of recovering the steady statebehavior associated with the use of a constant wall expansion ratio. We then conclude that thetime-dependent variation of the wall expansion ratio plays a secondary role that may be justifiablyignored. © 2010 American Institute of Physics. �doi:10.1063/1.3392770�

I. INTRODUCTION

Studies of laminar motions through porous channelscontinue to receive attention in the fluid and mathematicalcommunities due to their interesting connections with a va-riety of applications. These include binary gas diffusion, fil-tration, ablation cooling, surface sublimation, and the mod-eling of air circulation in the respiratory system. Berman1

initiated the analysis of the steady laminar flow of a viscousincompressible fluid in a two-dimensional porous channelwith uniform injection or suction. With the assumption thatthe transverse velocity component remained independent ofthe streamwise coordinate, Berman reduced the Navier–Stokes equations to a nonlinear ordinary differential equation�ODE� with appropriate boundary conditions. He then pro-ceeded to construct an asymptotic approximation for a smallReynolds number R using a regular perturbation scheme.

Extensions to Berman’s solution1 followed shortly there-after. For example, Terrill2 obtained series solutions for bothsmall and large values of R that compared favorably with thenumerical solutions to this problem. Using the method ofaverages, Morduchow3 presented a solution which coveredthe entire injection range. Later, White et al.4 derived apower series expansion that appeared to retain its validity forarbitrary R. However, their solution required the numericalcalculation of numerous power series coefficients that couldnot be tied recursively. Other studies devoted to the analysisof the symmetrically porous channel flow problem were re-ported by a diverse group of investigators. To list a few, onemay enumerate Sellars,5 Shrestha,6 Robinson,7 Skalak andWang,8 Brady,9 Durlofsky and Brady,10 Zaturska et al.,11

Taylor,12 and Yuan.13

Following this string of investigations, at least twoprominent categories of flow variations emerged, thus creat-ing new tracks for sustained research inquiry. The attendantmotions incorporated either asymmetries in the mean flowconfiguration or translational movement of the boundaries.The class of asymmetrical flows could be achieved, for ex-ample, by imposing different wall suction rates, as in thecase examined by Proudman14 under steady laminar condi-tions and large Reynolds numbers. The flow analog at small

a�Electronic mail: hangxu@sjtu.edu.cn.b�Electronic mail: sjliao@sjtu.edu.cn.c�Electronic mail: jzwu@coe.pku.edu.cn. Permanent address: State Key

Laboratory for Turbulence and Complex System, Peking University,Beijing 100871, China.

d�Author to whom correspondence should be addressed. Electronic mail:maji@utsi.edu.

PHYSICS OF FLUIDS 22, 053601 �2010�

1070-6631/2010/22�5�/053601/18/$30.00 © 2010 American Institute of Physics22, 053601-1

http://dx.doi.org/10.1063/1.3392770http://dx.doi.org/10.1063/1.3392770http://dx.doi.org/10.1063/1.3392770

R was equally investigated by Terrill and Shrestha.15

Shrestha and Terrill16 also provided accurate series solutionsfor large wall injection using the method of inner and outerexpansions.

The problem comprising moving boundaries appears tohave been initiated by Brady and Acrivos.17 These research-ers obtained an exact solution to the Navier–Stokesequations for the flow in a channel with an accelerating sur-face velocity. Their work was revisited in the context of amoving porous wall by Watson et al.18 Watson et al.19 thencombined asymmetry with wall movement in their analysisof the two-dimensional porous channel. Along similar lines,Dauenhauer and Majdalani20 presented a new similarity so-lution for the laminar, incompressible, and time-dependentNavier–Stokes equations in the context of a porous channelwith expanding or contracting walls. Assuming a constant �orquasiconstant� wall expansion ratio �, they reduced theNavier–Stokes equations into a self-similar ODE that couldbe solved numerically. Their approach employed Runge–Kutta integration coupled with a rapidly converging shootingtechnique to cover a modest range of R and wall expansionratios. Asymptotic solutions for the problem were later pro-vided by Majdalani and Zhou21 for moderate-to-large R andby Majdalani et al.22 for small R. The solution in the contextof rocket propulsion is also discussed by Zhou andMajdalani.23 As for the stability of these flows, this vitaltopic was originally addressed by Zaturska et al.,11 althoughseveral illuminating investigations have been carried out byMacGillivray and Lu,24 Lu,25 and Cox and King,26 to namea few.

In this paper, we revisit the porous channel flow problemwith regressing walls and solve it using a series expansionapproach known as the homotopy analysis method �HAM�.This method was developed by Liao27–31 as an improvementover the Adomian decomposition framework. In this context,it was developed for the purpose of obtaining series solutionsto strongly nonlinear differential equations. Its implementa-tion involves the introduction of an embedding parameter qthat permits converting the nonlinear governing equation intoa linear equation for q=0. As q is increased, homotopy en-sures that the original nonlinear equation is recovered in thelimit of q→1. As discussed in several articles,32–39 HAM hasbeen shown to exhibit several distinct advantages over otherasymptotic techniques. For example, HAM provides a con-venient control parameter � that helps to ensure seriesconvergence.28–31 HAM can also be applied with equal levelof ease to nonlinear ODEs and partial differential equations�PDEs�. This makes it an ideal approach for handling theDauenhauer–Majdalani PDE considered in this study.

To set the stage, we first revisit the exact similarity equa-tion rendered by Dauenhauer and Majdalani20 for the porouschannel flow. Given that these investigators were mainly in-terested in the type I injection and suction solutions thatpertained to their application, our analysis will seek to un-ravel not only the classical type I, but other possible solu-tions that have been classified as types II and III by Zaturskaet al.11 To this end, an optimal HAM-based approach will bedeveloped with the capability of capturing multiple solu-tions. This will enable us to obtain multiple solutions under

suction dominated conditions that have not been discussedbefore. By way of verification, our series approximationswill be shown to agree substantially well with the numeri-cally integrated solution described by Dauenhauer andMajdalani.20

In seeking further generalization, the Dauenhauer–Majdalani model20 will be extended to the case for which thewall expansion ratio � is no longer constant, but rather atime-dependent variable that transitions from �0 to �1. Thecorresponding similarity equation is readily turned into anonlinear PDE that can be solved analytically. While itwould be helpful to consider a case for which the channelhalf-height a varies from an initial value a0 to a final a1, sucha situation may be considered to be a special case of themodel used here. For example, knowing the initial and finalheights, one can calculate the average expansion ratio using,for example, ȧ��a1−a0� / �t1− t0�, where t1− t0 represents thetime for expansion. Then knowing ȧ, one can calculate �0= ȧa0 /� and �1= ȧa1 /�, where � is the kinematic viscosity.

The basic equations for the channel flow problem withboth constant and time-dependent wall expansion ratios aredescribed in Sec. II. In Sec. III, a conventional HAM ap-proach is implemented for the purpose of capturing one ofthe multiple solutions. This is followed by the presentation ofan optimal HAM-based approach that can be used to identifyany number of solutions: single, dual, or triple solutions forthe case of a constant wall expansion ratio. Therein, resultsare described and verified through comparisons to the nu-merical results obtained from the use of the Dauenhauer–Majdalani shooting approach.20 In Sec. IV, a HAM-basedapproach is advanced for the case of a time-dependent wallexpansion ratio. Finally, closing remarks and conclusions aregiven in Sec. V.

II. MATHEMATICAL DESCRIPTION

A. Basic equations

As shown in Fig. 1, the flow to be studied takes place ina two-dimensional rectangular channel bounded by porouswalls at y= �a. The walls of the channel undergo eitheruniform or nonuniform expansion or contraction in the trans-verse direction. Through the two opposing walls fluid is uni-formly injected or extracted at a constant speed vw. The gov-

x

y

0

a(t)

da/dtExpanding or Contracting Porous Wall

FIG. 1. Coordinate system and characteristic streamlines used to describethe fluid flow pattern.

053601-2 Xu et al. Phys. Fluids 22, 053601 �2010�

erning equations describing the unsteady flow of anincompressible fluid for mass and momentum conservationare

�u

�x+

�v�y

= 0, �1�

�u

�t+ u

�u

�x+ v

�u

�y= −

1

�

�p

�x+ �� �2u

�x2+

�2u

�y2� , �2�

�v�t

+ u�v�x

+ v�v�y

= −1

�

�p

�y+ �� �2v

�x2+

�2v�y2

� . �3�These are subject to the following boundary conditions:

u = 0, v = − vw; y = a�t� , �4�

�u

�y= 0, v = 0; y = 0, �5�

u = 0, v = 0; x = 0, �6�

where Cartesian coordinates �x ,y� that are fixed in space aretaken, with the x-axis extending along the length of the chan-nel and the y-axis in the wall-normal direction. Here u and vdenote the velocity components along x- and y-axes, and �,p, �, t, and vw correspond to the dimensional density, pres-sure, kinematic viscosity, time, and �crossflow� injection ve-locity at the wall. Note that Eqs. �4�–�6� are written in theupper half-domain �y�0�, thus taking into account symme-try with respect to the midsection plane while assuming anodd function for the wall normal velocity v and an evenfunction for the axial velocity u.

Using � to denote a stream function such that u=�� /�y and v=−�� /�x, the continuity equation is automati-cally satisfied and the vorticity transport equation is readilyobtained by taking the curl of the momentum equation. Onegets

��

�t+ u

��

�x+ v

��

�y= �� �2�

�x2+

�2�

�y2� , �7�

where

� =�v�x

−�u

�y= − �2� . �8�

Substituting the dual transformations20

� = ��x/a�F�,t�, = y/a �9�

into the vorticity transportation equation gives

�F + FF + F�2� − F� + �F − �a2/��Ft� = 0,�10�

which is subject to the four classical boundary conditions

� = 0: F = 0, F = 0, = 1: F = R , F = 0.

�11�Equation �10� describes the unsteady flow of an incompress-ible fluid in a porous channel. It is presented by White40 as

one of the new exact Navier–Stokes solutions attributed toDauenhauer and Majdalani.20 Here,

� = ȧa/� �12�

is the wall expansion ratio in which a and ȧ represent thehalf-height of the channel and its expansion speed; as before,� is the kinematic viscosity and so R=vwa /�=A� defines thecrossflow Reynolds number, with A being the injection coef-ficient. Note that according to our sign convention, a positiveR refers to injection conditions. More detail is given byDauenhauer and Majdalani20 where these steps are system-atically developed.

B. Equation for uniform wall expansion

When the wall expansion ratio � is constant in time, thefunction F becomes dependent on and � instead of � , t�.One can put Ft=0 and write

� = ȧa/� = ȧ0a0/� , �13�

where a0 and ȧ0 are the initial channel height and expansionrate, respectively. Integrating Eq. �13� with respect to t yields

a/a0 = �1 + 2��ta0−2�1/2. �14�

Moreover, given a constant injection coefficient A, one canput

ȧ/ȧ0 = vw�t�/vw�0� = �1 + 2��ta0−2�−1/2. �15�

As shown by Dauenhauer and Majdalani,20 an exact self-similarity equation emerges for Berman’s main characteristicfunction F. After some algebra, one retrieves

F + FF + F�3� − F� + �F = 0, �16�

with

= 0: F = 0, F = 0; = 1: F = R, F = 0,

�17�

where F denotes a sole function of . For more detail on themathematical foundations of Eqs. �16� and �17�, the reader isreferred to Dauenhauer and Majdalani.20

C. Equation for nonuniform wall expansion

Another practical setting corresponds to the case forwhich the wall expansion ratio � is not restricted to a con-stant, but is rather permitted to vary as a function of t,namely, from �0 to �1. Letting

�t� =a2

�, �18�

it follows that

��t� = ȧa/� = 12t�t� . �19�

Then taking into account Eq. �12�, we can put

�0�t� = ��0� = ȧ0a0/� =12t�0� . �20�

If we now consider the hypothetical case for which the in-jection coefficient A is inversely proportional to ��t�, Eq.�10� can be reduced to

053601-3 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

F + ��t��F + 3F� + FF − FF− �t�Ft = 0, �21�

where F=F� , t�. This PDE is subject to the same fundamen-tal boundary conditions delineated in Eq. �17�. Its formula-tion constitutes another exact reduction of the Navier–Stokesequations that will be discussed in Sec. IV.

III. SOLUTIONS FOR CONSTANT EXPANSION RATIOS

A. Conventional HAM approach

In this section, we provide an accurate analytical ap-proximation to Eq. �16�. With the help of HAM, the explicitapproximations for F�� can be specified as

F�� =

m=0

M

�m�� =

m=0

M

i=0

4m+4

�imai

mi, �22�

where �m�� is the so-called mth-order homotopy derivative,defined in the Appendix by Eq. �A13�, and the coefficientsai

m are given by

a1m = m�1

m−1a1m−1 + Cm

1 , �23�

a3m = m�3

m−1a3m−1 + Cm

3 , �24�

aim = �i

m−1aim−1

+��ei−4

m−1 + 3�ci−4m−1 + i−3di−5

m−1 + �i−4m + �i−4

m �i�i − 1��i − 2��i − 3�

�25�

for 3� i�4m+3, in which

bim = �i + 1��i+1

m ai+1m , �26�

cim = �i + 2��i + 1��i+2

m ai+2m , �27�

dim = �i + 3��i + 2��i + 1��i+3

m ai+3m , �28�

eim = �i + 4��i + 3��i + 2��i + 1��i+4

m ai+4m , �29�

�im =

k=0

m−1

r=max�0,i−4m+4k�

max�4k+4,i�

�rkar

kbi−rm−k−1, �30�

�im =

k=0

m−1

r=max�0,i−4m+4k�

max�4k+4,i�

brkci−r

m−k−1, �31�

C0m = C2

m = 0, �32�

C3m =

�

2 i=04m+4

ei−4m−1 + 3�ci−4

m−1 + i−3di−5m−1 + �i−4

m + �i−4m

i�i − 1��i − 2��i − 3�

−�

2 i=04m+4

ei−3m−1 + 3�ci−3

m−1 + i−2di−4m−1 + �i−3

m + �i−3m

i�i − 1��i − 2�,

�33�

C1m = −

i=0

4m+4ei−4

m−1 + 3�ci−4m−1 + i−3di−5

m−1 + �i−4m + �i−4

m

i�i − 1��i − 2��i − 3�

− C3m, �34�

m = � 0,m = 1,1,m � 1, �35�and

�im = �1, 1 � i � 4m + 3,

0, i � 1 or i � 4m + 3. �36�

Using the initial guess function given by

�0�,t� =12R �3 −

2� , �37�

the first four coefficients are found to be

a10 = 32R, a2

0 = 0, a30 = − 12R, a4

0 = 0. �38�

At this juncture, the explicit HAM series coefficients arefully determined. Other flow variables such as the velocity,pressure, and vorticity may be readily evaluated from theirbasic definitions. For example, the vorticity � may be re-trieved from Eq. �8�, viz.,

� = − �2� =�v�x

−�u

�y= − �xa−3F. �39�

Other variables are omitted here for the sake of brevity. Itmay be worth remarking that the convergence of the HAMapproximations is guaranteed when the linear operator L, theinitial guess �0��, and the values of the convergence-control parameter � are obtained in accordance with pre-established rules.28,29 Usually, L and �0�� are constructedbased on the solution expression consistent with the nonlin-ear behavior associated with the problem at hand. In this firstsection, we use �=−1. In a later section, we confirm thatseries convergence is guaranteed for ���− 32 ,−

23�, albeit not

limited to this range exclusively.In order to test the accuracy of our traditional HAM

series approximations with �=−1, we compare in Figs. 2�a�and 2�b� the HAM solution for the axial velocity, given byF��� /R, to the numerical results of Eq. �16�. The corre-sponding curves are evaluated for R= �5 and values of �ranging from �20 to 5. Clearly, a uniformly consistentagreement with numerics is realized owing, in large part, tothe ability of the HAM approximation to capture the true,nonlinear behavior of the solution in the limit of an infiniteseries.

For the suction driven channel with no wall motion,Zaturska et al.11 showed that one symmetric solution existsfor R� �−12.165,0� and three symmetric solutions emergewhen R� �−� ,−12.165�. In much the same way, the searchfor multiple solutions will have to be influenced by both Rand �, thus requiring a separate mathematical treatment, asfirst explained by Dauenhauer and Majdalani.20 This treat-ment will be carried out in Sec. III B. In the interim, thenumerical technique used by Dauenhauer and Majdalani20

will be used to study the emergence of solution multiplicityin suitable ranges of � and R. This is evident in Fig. 3 where

053601-4 Xu et al. Phys. Fluids 22, 053601 �2010�

one particular case is considered �e.g., �=1 and R=−20� forwhich three solution branches are seen to coexist. Using theconventional HAM approach, only type I can be predicted,although three types can be projected numerically. This ap-parent deficiency prompts us to modify the conventionalHAM approach to the extent of granting it more generality.As we describe in Sec. III B, the extension will enable us tocapture, using analytical series approximations, the threetypes of mean flow motions that are discussed by Zaturskaet al.11 in the case of a nonexpanding wall.

B. Optimal HAM approach for multiple solutions

In what follows, we describe the mathematical stepsneeded to mold the HAM to the porous channel flow prob-lem with a constant expansion ratio �. We thus considerthe steady PDE given by Eq. �16� and focus on its multiplesolutions.

Note that there exist dual or even triple solutions forsome values of R and �. In general, multiple solutions can be

elusive even by numerical methods. Here, we provide ananalytical framework that enables us to capture these mul-tiple solutions. First, we introduce the transformation

F�� = RS�� , �40�

and so, the original equation becomes

S� + ��S� + 3S�� + R�SS� − S�S�� = 0, �41�

subject to the boundary conditions

S�0� = 0, S��0� = 0, S�1� = 1, S��1� = 0, �42�

where the prime denotes a derivative with respect to . Tohelp identify the multiple solutions that accompany differentR and �, we define F��0�=R �, where

� = S��0� . �43�

Our strategy is guided by the shooting method describedearlier.20 Instead of using the original boundary conditions�42�, we first apply HAM to secure analytical approxima-tions for Eq. �41� using an initial value for S��0� at the lowerend of the domain. We hence take

S�0� = 0, S��0� = �, S��0� = 0, S�1� = 1, �44�

where � is not known initially. At the conclusion of theanalysis, � may be determined from the actual boundary con-dition that must be observed at the upper end of the domain,specifically

S��1� = 0. �45�

At this juncture, the form of the analytical solutions toEqs. �41� and �44� must be posited in terms of suitable func-tions. Given that Eq. �41� is defined over a finite domain� �0,1�, it is proper to express S�� in a power series of. Besides, using Eq. �41� and the boundary conditionsS�0�=0 and S�0�=0, it can be shown that

η

F’

(η)/

R

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

1.5

2

2.5

3

α = -20, -10, -5, 0, 1, 2, 2.5

(a)

η

F’

(η)/

R

-1 -0.5 0 0.5 10

0.5

1

1.5

2

α = -20, -10, -5, 0, 5

(b)

FIG. 2. Comparison between HAM-Padé approximations �circles� andnumerical results �solid lines� for �a� R=−5 and �b� R=5 over some valuesof �.

η

F’

(η)/

R

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

-3

-2

-1

0

1

2

3

4

5

Type IIIType IIType I

α = 1, R = -20

branch

FIG. 3. Three solutions for F��� /R at �=1 and R=−20.

053601-5 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

d2kSd2k

=0

= 0 when k = 1,2,3, . . . . �46�

Thus, S�� may be expressed as

S�� =

m=0

+�

Bm2m+1, �47�

where the undetermined coefficient Bm is dependent on �.Equation �47� defines what is often referred to as the solutionexpression for S��. According to the homotopy-based ap-proach, one needs to introduce an initial guess function ofthe form prescribed by Eq. �47� that still satisfies the prob-lem’s boundary conditions. For this reason, we choose, forthe case of a constant wall expansion ratio, the following oddpolynomial:

S0�� = A0 + A13 + A25. �48�

This initial guess function may be readily made to satisfy theconditions given by Eqs. �44� and �45�. The result is

A0 = �, A1 =52 − 2�, A2 = � −

32 , �49�

whence

S0�� = � + � 52 − 2��3 + �� − 32�5. �50�As with perturbation and traditional nonperturbation meth-ods, HAM transforms a nonlinear problem into an infinitenumber of linear subproblems. Additionally, HAM providesgreater freedom in selecting different base functions thatserve to approximate solutions more efficiently. Equally im-portantly, perhaps, HAM provides the means to ensure theconvergence of ensuing series solutions. This is accom-plished through the use of the convergence-control parameter�. By way of illustration, we note that, even for the nonlinearODE, Eq. �41�, it is not convenient to employ its entire linearpart L=S��+��S�+3S�� as the linear operator. Otherwise,the L=0 solution would contain the error function erf��; itwould then become exceedingly difficult to retrieve higher-order approximations. This obstacle can be overcome by se-lecting an auxiliary operator of equal order, in this case, offourth order, such as

L� =d4�

d4. �51�

This operator has the property

L�C0 + C1 + C22 + C33� = 0, �52�

where �C0 ,C1 ,C2 ,C3� are integral constants. This linear ap-proximation is sufficient to satisfy the rule of solution ex-pression, Eq. �47�. It should be noted that the solution forS��, where � �0,1�, represents a mapping operation ofS : �0,1�� �−� ,+��. Instead of solving Eq. �41� directly, weundergo continuous mapping according to

��;q�:�0,1� � �0,1� � �− �,+ �� , �53�

where

��;0� = S0��, ��;1� = S�� . �54�

The mapping operation follows from the zeroth-order defor-mation equation

�1 − q�L���;q� − S0��� = q�N���;q�� , �55�

which is subject to

��0;q� = 0, ���;q��

=0

= �,

�56�

�2��;q��2

=0

= 0, ��1;q� = 1.

On the one hand, q� �0,1� denotes the embedment param-eter mentioned before, while � alludes to the auxiliary pa-rameter known as the convergence-control parameter.27 Onthe other hand, L and N refer to the linear and nonlinearoperators based on Eqs. �51� and �41�, respectively. The lat-ter is defined by

N���;q�� =�4�

�4+ ���3�

�3+ 3

�2�

�2�

+ R���3��3

−��

�

�2�

�2� . �57�

To make further headway, one may evaluate Eq. �55� at bothends of its limiting expressions to deduce

�q = 0: L���;0� − S0��� = 0,q = 1: N���;1�� = 0; ∀ � � 0. �58�

In the above discussion, the initial guess S0��, defined byEq. �50�, satisfies the boundary conditions �44�. Hence, as wetrack the evolution of q across the range �0,1�, the solution ofthe zeroth-order deformation equation will vary continuouslyfrom the initial approximation S0�� to the target functionS�� that will exactly satisfy Eq. �44�. Expanding �� ;q� inthe Maclaurin series form with respect to q, we have

��;q� = S0�� +

m=1

+�

Sm��qm, �59�

where

Sm�� =1

m!

�m��;q�

�qm

q=0. �60�

Here, the relationship �� ;0�=S0�� is used. It should beemphasized that the zeroth-order deformation equation �41�contains the convergence-control parameter �, which is usedto guarantee the convergence of the final series solution. Inprinciple, � is chosen so that the above series remains con-vergent at q=1 over the entire domain, � �0,1�. Then, us-ing the relationship �� ;1�=S��, we obtain the homotopy-series solution

053601-6 Xu et al. Phys. Fluids 22, 053601 �2010�

S�� = S0�� +

m=1

+�

Sm�� , �61�

where Sm�� is unknown. Its governing equation and relatedboundary conditions have to be carefully established. To thisend, we introduce the vector

S�m = �S0��,S1��,S2��, . . . ,Sm��� . �62�

Differentiating the zeroth-order deformation equation �55�and the boundary conditions �56� m times with respect to q,dividing by m!, and then setting q=0, we obtain the so-calledmth-order deformation equation

L�Sm�� − mSm−1��� = ��m�Sm−1��� . �63�

Equation �63� is subject to

= 0: Sm = 0, Sm� = 0, Sm� = 0; = 1: Sm = 0,

�64�

where

m = �0, m � 11, m � 1, �65�and the right-hand-side function �m is given by

�m�Sm−1��� = Sm−1�� + ��3Sm−1� + Sm−1� �

+ R

n=0

m−1

�SnSm−1−n� + Sn�Sm−1−n� � . �66�

According to Eq. �51�, it is straightforward to retrieve a par-ticular solution Sm

� �� of Eq. �63� by integrating the right-hand-side term four times with respect to , viz.

Sm� �� = mSm−1��

+ �� � � � �m�S�m−1,�dddd . �67�The ensuing quadfold integration yields

Sm�� = Sm� �� + Cm,0 + Cm,1 + Cm,22 + Cm,33 �m � 1� ,

�68�

where the integral constants

Cm,0 = Cm,2 = 0, Cm,1 = − dSm�d =0, and�69�

Cm,3 = − Cm,1 − Sm� �1�

are secured from the boundary conditions in Eq. �64�. TheMth-order approximation of S�� may hence be written as

S�� � S0�� +

k=1

M

Sk�� . �70�

Note that the above expression incorporates the unknownparameter �.

The next step is to determine the value of � that willpermit the solution to satisfy the boundary condition �45� at

the upper end of the domain, namely, S��1�=0. Substitutingthe Mth-order approximation �70� into Eq. �45� gives

�M��,�� = S0��1� +

n=1

M

Sn��1� = 0, �71�

where �M represents the expanded form of the constrainedboundary condition. Using an Mth-order approximation, wehave

SM�� =

n=0

3M+2

AM,n��,�,��2n+1 �72�

and so

�M��,�� =

n=0

M+1

BM,n��,���n = 0, �73�

where AM,n�� ,� ,�� and BM,n�� ,�� are coefficients of thepower series of . Note that, as long as � is given, the solu-tions of Eq. �71� may be readily obtained.

In our quest for an optimal value of �, we use a tech-nique that has been shown to produce a fast convergingHAM approximation. In principle, the technique seeks tominimize the exact square residual error of Eq. �41� at themth-order. This quantity is given by

Êm��,�� = �0

1 �N�

n=0

m

Sn����2d . �74�In practice, however, the evaluation of Êm�� ,�� tends to betime-consuming. A simpler alternative consists of calculatingthe averaged square residual error. In this vein, one may seekto minimize the zeroth-order discretization of Eq. �74�,

Em��,�� =1

�M + 1�k=0M �N�

n=0

m

Sn�k���2, �75�where

k = k� =k

M, k = 0,1,2,3, . . . ,M . �76�

Hereafter, a value of M =40 will be used with the purpose ofoptimization. For traditional problems in which Em is onlydependent on �, it is relatively straightforward to seek anoptimal value �� for which Em is minimized. However, forthe problem under consideration, the residual error is exac-erbated by its dependence on both � and �. In fact, bothEm�� ,�� and �M�� ,�� contain the two unknowns: � and �.So, while the optimal convergence-control parameter �� canstill be determined from the minimum of Em�� ,��, it must beadditionally subjected to the algebraic equation �71� neededto secure the boundary value constant �. For this doublycoupled optimization problem, we let ��� ,��� denote the so-lution of the algebraic equation �71�, which corresponds tothe minimum of the square residual error. Mathematically, itholds that

053601-7 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

���,��� = min�Em��,��,�m��,�� = 0� . �77�

For the sake of computational efficiency, the exact squareresidual error may be replaced by the averaged square re-sidual error �75�. To identify the optimal approximation in aregion a���b, we take

���,��� = min�Em��,��,�m��,�� = 0,a � � � b� . �78�

Such an operation may be relegated to symbolic program-ming with a dedicated “Minimize” command.

Before leaving this section, it may be helpful to remarkthat Eq. �63�, the high-order deformation equation representsa linear ODE that is relatively simple to solve. This behavioris contrary to that of Eq. �41�, the original nonlinear ODE foruniform wall expansion with �=const. Using homotopy, theDauenhauer–Majdalani ODE is transferable into an infinitenumber of linear ODEs that can be solved directly. The en-suing transformations are made possible by the flexibility ofthe HAM procedure in identifying a simple auxiliary linearoperator through which the solution of the high-order defor-mation equation can be readily obtained. Meanwhile, the so-called convergence-control parameter � may be optimized tothe extent of accelerating or ensuring the convergence of theseries approximations.28–31,41 These particular characteristicsrepresent some of the key features that distinguish HAMfrom other analytical methods.

C. Multiple solutions

For constant �, Dauenhauer and Majdalani20 obtained anexact ODE and provided the steps that can lead to a numeri-cal solution using a fast-converging shooting methodcoupled with Runge–Kutta integration. They illustrated theirresults by providing numerical solutions over a modest rangeof R and � that are aligned with the type I family of solutionsclassified by Zaturska et al.11 However, solutions of types IIand III for the suction case, particularly those leading tointernal flow steepening and recirculation, were alluded tobut not considered in their article. Later studies by Majdalaniand Zhou,21 Majdalani et al.,22 and Zhou and Majdalani23

provided closed-form explicit approximations over somelimited ranges of suction and injection.

To illustrate how the general HAM approach maybe applied, we first consider a special case of �=4 andR=−10. We can sequentially solve up to the mth-order de-formation equations �63� and �64� for m=1,2 ,3 , . . .. Wehence retrieve

S1�� = ���277 + 3173�6930 − 325924�3 − ��3�5 − 12�5+ ��2�2

21−

�

7+

1

28�7

− ��10�263

−55�

126+

25

84�9

+ ��5�299

−5�

33+

5

44�11, �79�

S2�� = S1�� + �2� 63 355�36 432 426 − 15 323 311�2

192 972 780

+159 557 617�

643 242 600−

396 889

2 450 448�3 + ¯ . �80�

At the mth-order of approximation, Sm�� contains the un-known convergence-control parameter � and the unknownvariable � that satisfies the right-hand-side boundary condi-tion connected with Eq. �71�. At the first- and second-orderapproximations, Eq. �71� yields

�1 = �� 41154 − 421�1155 − 116�2

693� = 0 �81�

and

�2 = ��4177 − 842�1155 − 232�2

693� + �2� 3215

37 128

−24 693 629�

107 207 100+

773 027�2

6 432 426−

2033�3

153 153� = 0, �82�

respectively, up to �m. Thus, the algebraic equation�m�� ,��=0 contains not only the unknown variable � butalso the unknown convergence-control parameter �. Eachzero �� of �m��� ,��=0 will hence signal the presence of adistinct solution. After determining the number of zeros, aproper choice of � can be sought for each ��. It can thus beseen that the convergence-control parameter � has no physi-cal meaning besides granting us the freedom to calibrate itsvalue to the extent of optimizing the solution. For example,in case of the tenth-order approximation for R=−10 and�=4, the curves of �10�� ,�� for �=−

32 ,−1 and −

23 are com-

puted and illustrated in Fig. 4. It is interesting that, although�10�� ,�� curves are sensitive to the value taken by �, thezeros of the algebraic equation �10�� ,��=0 when ���− 32 ,− 23� remain close. They actually become virtually indiscern-ible as m is increased. The zeros appear to be only weaklysensitive to � despite the existence of two zeros in the re-

γ

Γ 10

-3 -2 -1 0 1 2 3

-4

-3

-2

-1

0

1

2

3

4

γ2nd branch γ1st branch

FIG. 4. Curves of �m�� ,�� using a 10th-order HAM approximation�m=10� in case of R=−10 and �=4. Solid line: �=−1; dashed line: �=− 32 ;dash-dotted line: �=− 23 .

053601-8 Xu et al. Phys. Fluids 22, 053601 �2010�

gions �� �−2,−1� and �� �0,1�. Each of these zeros corre-sponds to a distinct solution.

At the given order of approximation m, the optimalconvergence-control parameter �� may be determined for thepositive solution of �� �0,1� by minimizing the averagedsquare residual error, specifically

���,��� = min�Em��,��,�m��,�� = 0,� � 0� . �83�

The first and second sets of outcomes obtained for �� and ��

through minimization are cataloged in Table I at differentapproximation orders. It is clear that as the order of approxi-mation increases, the optimal convergence-control parameter�� tends to �1.766 for the first solution, while the attendant�� approaches a positive value of approximately 0.625 00. Itmay be seen that Em��� ,���, the averaged square residualerror entailed in this analysis, decreases monotonically as mis increased. For the second solution, �� tends to �1.5 while�� approaches a negative value of �1.189 79. Here, too, theaveraged square residual error Em��� ,��� decreases mono-tonically as well.

For R=−10 and �=4, the analytical approximations ofF�0� /R and F��0� /R are listed in Table II for the twosolutions at hand. While the set F��0� /R�0.625 007and F��0� /R�8.255 45 corresponds to the first solution, asecond set is obtained for the second solution with F��0� /R

�−1.190 33 and F��0� /R�35.8542. As the order of ap-proximation is increased, the averaged square residual erroris seen to decrease monotonically down to 7.99�10−8; in themeantime, F��0� /R and F��0� /R asymptotically converge tofixed values.

As shown in Fig. 4, there also exists a negative solutionfor �� �−2,−1�. At the prescribed order of approximation m,the companion parameter �� may be determined from

���,��� = min�Em��,��,�m��,�� = 0,� � 0� . �84�

The convergence ratio of the homotopy-series solution canbe accelerated by means of the so-called homotopy-Padétechnique.27–31 As shown in Table III, the homotopy-Padétechnique enables us to identify the convergent approxima-tion F��0� /R�0.625 007 396 and �1.190 323 for the firstand second solutions, respectively. These predictionsagree quite well with the numerical values of 0.625 007 4and �1.190 323 reported for these cases. Note that ouranalytical projections agree well with those reproduced usingDauenhauer and Majdalani’s numerical approach.20 Theseare shown in Fig. 5 where the axial flow velocity is illus-trated using the numerical solution alongside a 20th-orderHAM approximation. The two featured solutions are ac-quired using �=− 74 ,

32 at a crossflow Reynolds number of

R=−10 and a wall expansion ratio of �=4. In short, thisexample illustrates how the homotopy-based approach can

TABLE I. The optimal convergence-control parameter �� and the corre-sponding averaged square residual error for the first ���0� and secondsolutions ���0� in the case of R=−10 and �=4.

m, orderof approximation �� �� Em��� ,���

First solution 1 �0.886 0.577 336 32.33

5 �1.022 0.609 118 5.25�10−2

10 �1.727 0.624 117 1.37�10−3

15 �1.764 0.624 840 8.07�10−5

20 �1.770 0.624 963 6.11�10−6

Second solution 5 �1.303 �1.236 36 2.40

10 �1.479 �1.190 69 0.11

15 �1.515 �1.188 51 2.56�10−2

20 �1.458 �1.189 79 2.26�10−4

TABLE II. The mth-order approximation of the first ���0� and second solutions ���0� in the case ofR=−10 and �=4.

�m, order

of approximation F��0� /R F��0� /R Em�� ,��

First solution − 74 10 0.624 161 8.267 56 1.51�10−3

20 0.624 967 8.256 03 6.55�10−6

30 0.625 005 8.255 48 3.68�10−8

40 0.625 007 8.255 45 1.81�10−10

50 0.625 007 8.255 45 1.11�10−12

Second solution − 32 10 �1.189 95 35.8984 0.150

20 �1.190 03 35.8474 6.54�10−4

30 �1.190 41 35.8555 1.05�10−5

40 �1.190 31 35.8539 8.46�10−7

50 �1.190 33 35.8542 7.99�10−8

TABLE III. The �m ,m� homotopy-Padé approximation of the first ���0�and second solutions ���0� in the case of R=−10 and �=4.

m

First solution Second solution

F��0� /R F��0� /R F��0� /R F��0� /R

4 0.624 478 732 8.265 444 222 �1.219 891 36.010 91

8 0.625 005 336 8.255 477 422 �1.178 465 35.178 78

12 0.625 007 412 8.255 446 248 �1.190 349 35.854 62

16 0.625 007 395 8.255 446 127 �1.190 319 35.854 09

18 0.625 007 396 8.255 446 127 �1.190 319 35.854 11

20 0.625 007 396 8.255 446 125 �1.190 323 35.854 15

22 0.625 007 396 8.255 446 124 �1.190 323 35.854 16

24 0.625 007 396 8.255 446 124 �1.190 323 35.854 16

053601-9 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

indeed capture the dual solutions of the nonlinear boundary-value problem for channel flows with constant wall expan-sion.

The present approach is capable of capturing triple rootswhen three branches of solution arise. A concrete exampleconsists of the case of R=−11 and �=1.5. The correspondingcurves of �10�� ,�� at the tenth order of approximation aregiven in Fig. 6 using �=−1, − 32 , and −

23 . This graph clearly

shows that two values of � emerge near �1 and 0. Besides,it appears that a third solution exists in the region of ��1. Inthe case of R=−11 and �=1.5, Eq. �71� produces three so-lutions near �1, 0, and ��1. In seeking an optimalconvergence-control parameter, we carefully bracket oursearches within the following intervals:

�A�: min�Em��,��, �m��,�� = 0, − 2 � � � − 0.5� ,

�85�

�B�: min�Em��,��, �m��,�� = 0, − 0.5 � � � 0.5� ,

�86�

�C�: min�Em��,��, �m��,�� = 0, 0.5 � � � 4� . �87�

Forthwith, the results of our analysis are posted in Table IV.We find �� to be approximately − 53 for the first and secondsolutions, and − 45 for the third. The HAM approximationsof F��0� /R and F��0� /R associated with these three solu-tions are listed in Table V. Using the homotopy-Padé tech-nique, we obtain F��0� /R=−1.023 771 2 and 0.169 352 forthe first and second solutions, respectively. These valuesagree very well with the numerical predictions of F��0� /R=−1.023 771 and 0.169 353 2 for these two branches. Incontrast, we find the third solution to be more difficult toretrieve. This may be attributed to its averaged square re-sidual error remaining 5.68�103 at the tenth-order approxi-mation and slowly decreasing to 1.08�10−3 at the 60th order�see Table VI�. Everywhere, the three analytical approxima-tions of F��� /R appear to be in excellent agreement withthe numerical predictions shown in Fig. 7. This uniformlyconsistent agreement with numerics is realized owing, inlarge part, to the ability of the HAM approximation to cap-ture the true, nonlinear behavior of the solution in the limitof an infinite series.

Generally speaking, the numerical identification of allpossible solutions associated with nonlinear boundary-value problems can require significant effort. For thesuction driven channel flow with no wall motion, Zaturskaet al.11 showed that one symmetric solution exists forR� �−12.165,0� and three symmetric solutions emerge whenR� �−� ,−12.165�. In the present work, the same types ofsolutions are captured. Using HAM, dual solutions are re-turned, for example, in the case of R=−10 and �=4, whilethree distinct solutions are retrieved in the case of R=−11and �=1.5. The accuracy of our analytical predictions may

η

F’

(η)/

R

-1 -0.5 0 0.5 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

FIG. 5. Comparison of numerical solutions with the 20th-order HAM ap-proximation for F��� /R in case of R=−10 and �=4. Symbols: numericalresults; dashed line: first solution with �=− 74 ; solid line: second solutionwith �=− 32 .

γ

Γ 10

-4 -3 -2 -1 0 1 2 3 4

-4

-2

0

2

4

γ1st solution

γ3rd solutionγ2nd solution

FIG. 6. Curves of �m�� ,�� using a tenth-order HAM approximation�m=10� in case of R=−11 and �=1.5. Solid line: �=−1; dashed line:�=− 32 ; dash-dotted line: �=−

23 .

η

F’

(η)/

R

-1 -0.5 0 0.5 1-3

-2

-1

0

1

2

3

R = -11, α = 1.5Lines: analytic approximations given by the HAMSymbols: numerical results

FIG. 7. Comparison of numerical solutions with the 30th-order HAM ap-proximation of F��� /R in case of R=−11 and �= 32 . Symbols: numericalresults; solid line: first solution with �=− 53 ; dashed line: second solutionwith �=− 53 ; dash-dotted line: third solution with �=−

34 .

053601-10 Xu et al. Phys. Fluids 22, 053601 �2010�

TABLE IV. The optimal convergence-control parameter �� and the corresponding averaged square residualerror in the case of R=−11 and �=1.5 for three solutions of ��.

m, orderof approximation �� �� Em��� ,���

First solution 4 �1.235 �0.982 73 11.9

8 �1.604 �1.020 08 0.153

12 �1.561 �1.022 75 2.37�10−2

16 �1.654 �1.023 56 2.49�10−4

20 �1.625 �1.023 71 1.16�10−4

Second solution 6 �1.661 0.212 76 24.6

8 �1.701 0.176 35 2.18

12 �1.441 0.166 88 2.40�10−2

16 �1.146 0.167 17 1.69�10−2

20 �1.681 0.169 45 5.95�10−4

Third solution 20 �0.951 2.753 98 17.3

24 �0.859 2.757 36 12.2

26 �0.940 2.750 98 2.70

28 �0.881 2.759 36 1.09

30 �0.875 2.766 89 0.37

TABLE V. The mth-order approximation of the three solutions in the case of R=−11 and �= 32 .

�m, order

of approximation F��0� /R F��0� /R Em�� ,��

First solution − 53 10 �1.022 322 24.270 70 0.468

20 �1.023 716 24.285 64 1.52�10−4

30 �1.023 769 24.286 28 1.60�10−7

40 �1.023 771 24.286 31 5.01�10−10

50 �1.023 771 24.286 31 1.72�10−12

Second solution − 53 10 0.166 208 10.2842 0.67

20 0.169 384 10.2447 7.20�10−4

30 0.169 378 10.2449 2.19�10−5

40 0.169 335 10.2452 9.98�10−7

50 0.169 354 10.2451 3.49�10−8

Third solution − 34 20 2.779 59 �15.6613 3.62�10+2

30 2.762 18 �15.5188 1.49�10+1

40 2.761 14 �15.5128 0.30

50 2.761 24 �15.5134 7.40�10−3

60 2.761 11 �15.5122 1.08�10−3

TABLE VI. The �m ,m� homotopy-Padé approximation of three solutions in the case of R=−11 and �= 32 .

m

First solution Second solution Third solution

F��0� /R F��0� /R F��0� /R F��0� /R F��0� /R F��0� /R

4 �1.023 162 1 24.292 585 1 0.166 859 0 10.282 860 ¯ ¯

8 �1.023 770 0 24.286 379 7 0.167 998 0 10.239 451 2.815 91 �15.8950

12 �1.023 771 1 24.286 308 6 0.169 360 1 10.245 072 2.762 90 �15.5284

16 �1.023 771 2 24.286 308 8 0.169 351 8 10.245 026 2.761 54 �15.5157

20 �1.023 771 2 24.286 308 8 0.169 353 2 10.245 150 2.761 13 �15.5123

22 �1.023 771 2 24.286 308 8 0.169 353 1 10.245 152 2.761 11 �15.5123

24 �1.023 771 2 24.286 308 8 0.169 353 2 10.245 151 2.761 11 �15.5123

053601-11 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

hence be viewed as supportive of the potential application ofHAM as a guide, if not alternative, to numerical solutions ofnonlinear boundary-value problems, particularly, of thosearising in fluid mechanics.

It is clear that our solutions correspond to the three typesof mean flow profiles discussed by Zaturska et al.11 in thecase of a stationary porous wall. Due to the striking similari-ties between our analytical solutions and theirs, we have la-beled our branches after theirs. This behavior is expectedbecause the problem related to the porous channel flow withstationary walls should be recoverable from our analysis inthe limiting case of �=0.

To showcase the flow behavior corresponding to the dif-ferent branches of solution, fluid streamlines are plotted inFig. 8 for �=1.5 and R=−11. This graph depicts all threetypes of solutions and enables us to deduce their fundamentalcharacteristics under conditions conducive of wall suction.

Overall, both types I and II share a similar characteristic;it is possible for the flow drawn at the porous wall to origi-nate from the chamber volume bounded anywhere between0�r�1. By way of contrast, it may be seen that the type IIprofile undergoes faster flow turning above the porous wall,as corroborated by the sharper streamline curvatures depictedin Fig. 8. For the type III profile, the fluid removed at thesurface may only originate from an annular region that ex-tends over ��r�1. In the case of a nonexpanding wall��=0�, an expression for the inner distance � of the type IIIannulus may be determined using a transcendental relationthat can be obtained asymptotically for R→−�. As shown byLu,42 this relation is given by

�R��exp�− R�� = −R8

2�9 exp�1�, �88�

where R is negative for wall suction. Lu’s type III solutionemerges asymptotically and may be described by the com-pact trigonometric function,

F�y� = −1 − �

��sin� �y

1 − �� . �89�

At this juncture, it may be useful to remark that the curvaturedisparity between the first two types will gradually vanishwith increasing �R�. This is due to the convergence of the twobranches onto a single polynomial expression that is elabo-rately discussed by Robinson7 and Zaturska et al.11 Accord-ingly, the type I and type II branches collapse into the essen-tially irrotational form,

F�y� = y + O�R−1�; R → − � . �90�

Along similar lines, only one solution is confirmed for smallsuction with R� �−12.165,0�. This is given by

F�y� = 12 y�3 − y2�; R → 0−. �91�

When attention is turned to injection dominated conditions,the unique branch for the small suction case continues tohold as the crossflow Reynolds number undergoes a signswitch. Two asymptotic solutions are known to date andthese correspond to either small or large injection Reynoldsnumbers. According to Berman,1 the small injection meanflow solution remains identical to the one obtained for smallsuction, namely,

F�y� = 12 y�3 − y2�; R → 0+. �92�

However, as the crossflow Reynolds number is increased, atrigonometric solution emerges, specifically

F�y� = sin� 12�� ; R → + � . �93�Equation �93� for the large injection case has often beentermed “Taylor’s profile” due to its relevance to several tech-nological applications that encompass paper manufactureand both solid and hybrid propellant gas dynamics.

Before leaving this subject, it may be instructive to re-mark that, at first, we have initiated our analysis by exploringthe type I HAM solutions for which F�y�= 12 y�3−y

2� repre-sents a valid initial approximation for the problem targetedby Dauenhauer and Majdalani.20 We have then extended ouranalysis by uncovering the additional types II and III withoutchanging the initial guess function. We have thus signifi-cantly broadened the range of applicability of analytical rep-resentations for the types II and III branches beyond thoseachieved asymptotically in previous studies. These includeseveral asymptotic solutions developed by Majdalani andco-workers21–23 over some special ranges of R and ��0. Itmust be borne in mind, however, that the types II and IIIbranches of solutions are expected to be mostly unstableaccording to the projections made by Zaturska et al.,11 Luet al.,25 and MacGillivray and Lu.24

As we explore wider ranges of � and R using ourHAM-based approach, we find that when −10.727 376�R�−8.704 297 0, dual solutions are possible for some valuesof �. In this situation, when the values of F��0� are plottedversus � in a range of R, a closed circle is formed, as shownin Fig. 9. Our analytical solutions agree with the numericalresults given by the shooting integration method. Specifi-cally, the value of F��0� shown on the graph corresponds to

ν x /a

η

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1R = -11, α = 1.5

FIG. 8. Streamline patterns of three types of analytical solutions in case ofR=−11 and �= 32 . Solid line: type I solution with F��0� /R=2.761 11; dashedline: type II solution with F��0� /R=0.169 353 2; dash-dotted line: type IIIsolution with F��0� /R=−1.023 771 190.

053601-12 Xu et al. Phys. Fluids 22, 053601 �2010�

the axial velocity evaluated along the channel’s midsectionplane. The presence of dual or triple solutions will thus leadto dissimilar mean flow profiles.

IV. SOLUTIONS FOR TIME-DEPENDENT WALLEXPANSION RATIOS

The analysis so far has been centered on the assumptionthat � remains constant during the expanding or contractingmotions of the porous wall. In what follows, we relax thiscondition and allow the wall expansion ratio to become timevariant. This enables us to test the validity of the originalassumption leading to the exact Navier–Stokes reduction.Our choice of ��t� is guided by the formulations presentedthrough Eqs. �18�–�21�. For example, by taking

��t� = �0 exp�− t� + �1�1 − exp�− t�� , �94�

we obtain

�t� = − 2�0 exp�− t� + 2�1t + 2�1 exp�− t� . �95�

It can thus be seen that the wall expansion ratio ��t� variesexponentially. If we further assume that �0 and �1 representthe initial and final values of ��t�, we can put

��t� = �1 + ��1 − �0�t

1 + t, �96�

whence

�t� = 2�1t + 2��0 − �1�log�1 + t� . �97�

In what follows, we find it useful to employ the algebraicapproximations �96� and �97� in lieu of the exponentiallydecaying forms given by Eqs. �94� and �95�. In the presentcase, we solve a nonlinear PDE, here Eq. �21�, with theboundary conditions given by Eq. �17�. In general, a nonlin-ear PDE is substantially more challenging to solve than anODE, even by means of numerical techniques. Besides,PDEs are traditionally regarded as fundamentally differentfrom ODEs. So the question that comes to mind is this: Can

we transform a nonlinear PDE into an infinite number oflinear ODEs? The answer is positive: in many cases,30 anonlinear PDE can be indeed broken down into a sequenceof linear ODEs by means of homotopy.41 The analyticaltreatment of a nonlinear, unsteady PDE with a time-dependent wall expansion ratio ��t� is rather similar to thatemployed in the treatment of a nonlinear ODE with constant�. The procedure is straightforward and so, in the interest ofbrevity, the steps are outlined below and further detailed inthe Appendix. First, F� , t� is expressed by a series

F�,t� = F0�,t� +

m=1

+�

Fm�,t� , �98�

where Fm� , t� is controlled by the high-order deformationequation

L�Fm�,t� − mFm−1�,t�� = ��̂m�,t� �99�

with

Fm�0,t� = 0,�100�

�2Fm�2

=0

= 0, Fm�1,t� = 0, �Fm� =1 = 0,in which

�̂m�,t� = Fm−1� + ��t��3Fm−1� + Fm−1� � − �t�Ṡm−1�

+

n=0

m−1

�FnFm−1−n� + Fn�Fm−i−n� � . �101�

Here, the primes and dots denote differentiation with respectto and t, respectively, the auxiliary linear operator L isgiven by Eq. �A5�, and m is defined by Eq. �65�. For sim-plicity, we choose a simple initial guess

F0�,t� =12R�3 −

2� , �102�

which satisfies Eq. �17�. Subsequently, the nonlinear PDEdefined by Eq. �21� may be converted into an infinite numberof linear ODEs using HAM, as shown in the Appendix.

The sensitivity of the solution with respect to the choiceof a time-dependent model for ��t� is illustrated in Fig. 10.Therein, the axial velocity profile is plotted at R=1 and sev-eral instants of time that correspond to a variation in ��t�from �2 to �1. It is interesting to note the general agree-ment between the two families of curves. As one may expect,the profiles using the exponentially decaying behavior for��t� vary slightly more rapidly than those based on the alge-braic approximation. Another case is illustrated in Fig. 11where the axial velocity profile F�0, t� is plotted as a func-tion of t while ��t� is varied “to and fro” between 1 and 2. Itis clear that the velocity decays monotonically when ��t� isreduced from 2 to 1; the converse is true as ��t� is incre-mented from 1 to 2. The significant outcome to report here isthat, regardless of direction taken, the velocity profilesquickly approach the steady state conditions. Similar trendsare displayed by other flow variables. This rapid shift tosteady state behavior helps to confirm the validity of theanalysis performed heretofore, specifically in Sec. III where

α

F’

(0)

0 2 4 6 8 10 12 14-10

-5

0

5

10

15

20R = -8.72, -8.8, -9, -9.5, -10, -10.5

FIG. 9. Multiple solutions for varying R and � �suction conditions�. Sym-bols: analytical results given by the HAM-based approach; line: numericalresults given by the shooting method.

053601-13 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

the time dependence is deliberately ignored as a requirementto reduce the Navier–Stokes equations into a single, fourthorder ODE.

To illustrate the time-dependent evolution of the axialvelocity, Figs. 12�a� and 12�b� are used to depict the behaviorof F� at two different injection coefficients, specifically, forA= �5. These results correspond to the direct temporal so-lution of Eq. �21�. To verify the rapid shift to steady stateconditions, the axial velocity profiles are shown fort=0,0.5,1.0,2.0,5.0,10.0. Note that, at the scale used inthese graphs, the curves beyond t=5 cannot be visually dis-tinguished. Along similar lines, the spatial variation in thenormal velocity is illustrated in Figs. 13 and 14. In Fig. 13,the sensitivity of the normal velocity is illustrated at threedifferent wall expansion ratios, �=−5,0 ,+5, hence includingthose caused by suction, stationary motion, or injection.These are computed using a 40th-order HAM approximationand a fixed crossflow Reynolds number of R=5. As for Fig.

14, it illustrates the time evolution of the normal velocity fort=0,0.5,1.0,2.0,5.0, an injection coefficient of A=5, and atime-dependent wall expansion ration that varies from �1 to�2. Here, too, results beyond t=5.0 become indiscernible,thus signaling the onset of steady state conditions.

V. CONCLUSIONS

In this study, the porous channel with orthogonally mov-ing walls is revisited in the context of laminar incompress-ible motion with both uniform and nonuniform wall regres-sions. The flow in the case of uniform wall regression isdescribed by a nonlinear ODE, but the flow in the case ofnonuniform wall regression is prescribed by a nonlinearPDE. For each case, an analytic approach based on the HAMis presented, thus transforming the original nonlinear ODE orPDE into an infinite number of linear ODEs whose solutionscan be obtained by symbolic computation software. At theoutset, a recursive series formulation is derived that can bereadily evaluated using generic programming. The HAM-based procedure may thus be viewed as a technique that can

η

Fη(

η,t)

0 0.1 0.2 0.3 0.4

1.3

1.4

1.5

1.6

1.7

t =0,

0.25

,1.0

,5

exponential

algebraic

FIG. 10. Sensitivity of the axial velocity F� , t� at R=1 as ��t� is variedfrom �0=−2 to �1=−1.

t

Fη(

0,t)

0 1 2 3 4 5 6 7 8 9 10-1.65

-1.6

-1.55

-1.5

α0=1, α1=2

α0=2, α1=1

FIG. 11. Temporal variation in the centerline velocity F�0, t� for R=1.

η

F’

(η,t

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

-7

-6

-5

-4

-3

-2

-1

0

t = 5.0, 10.0

t = 0.0

t = 0.5

t = 2.0

t = 1.0

Solid line: t = 5.0Dash dotted line with symbols: t = 10.0

(a)

η

F’

(η,t

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

t = 5.0, 10.0

t = 0.0

t = 0.5

t = 1.0

t = 2.0

Solid line: t = 5.0Dash dotted line with symbols: t = 10.0

(b)

FIG. 12. Spatial variation in the axial velocity F� , t� for some values of tusing �a� A=−5 and �b� A=5.

053601-14 Xu et al. Phys. Fluids 22, 053601 �2010�

greatly simplify the solution of nonlinear ODEs or PDEs. Inthis endeavor, multiple solutions may be connected to a finitenumber of distinct zeros, here called ��, of a constraint equa-tion associated with the shooting approach. After identifyingthe distinct solutions associated with each zero, aconvergence-control parameter � may be carefully selectedto ensure the convergence of the series at hand. In the presentwork, an error minimization approach is employed to deter-mine an optimal value �� that leads to fast series conver-gence. When necessary, the homotopy-Padé technique is in-voked to further accelerate series convergence. At length, aseries approximation is obtained for the nonlinearDauenhauer–Majdalani ODE, in the case of constant wallexpansion, and for the time-dependent PDE, in the case of atime-varying wall expansion.

For the uniform wall expansion ratio, our HAM-basedapproach is shown to capture dual and triple solutions indifferent ranges of R and �. We not only recover the type Ibranch of solutions pursued by Dauenhauer and Majdalani,20

but also capture the type II and type III families reported byZaturska et al.11 These solutions were alluded to although notpursued in Dauenhauer and Majdalani’s former article.20 Thenew profiles involve either steeper flow turning at the wall�type II� or annular flow splitting and recirculation that arethoroughly described by Lu �type III�. The high-order ap-proximations presented here to describe all three categoriesof motion are shown to exhibit substantial agreement withthe numerical results acquired using the integration algo-rithm constructed by Dauenhauer and Majdalani.20

For the nonuniform wall expansion case, two models for��t� are chosen that involve either exponential or algebraicvariation from an initial �0 to a final �1. With this assump-tion in effect, the Navier–Stokes equations are transformedinto a PDE using a judicious choice of similarity variables.The resulting problem is solved analytically by HAM andshown to quickly reproduce the steady state solution ob-tained previously. This is due to the rapid decay of the time-dependent transients and the swift convergence of the solu-tion to conditions corresponding to a constant wall expansionratio.

Finally, it is clear that HAM offers several advantageswhen compared with other methods of solution. First, whilethe reduced ODE has been previously solved using asymp-totics, its accuracy has remained dependent on the size of thesmall perturbation parameter 1 /R. Unlike the perturbationapproximation that applies to a limited range of Reynoldsnumbers, the HAM solution has no limitations. It is not onlyindependent of the size of 1 /R but it also provides, as awindfall, a convenient tool that can be calibrated to ensureseries convergence. Second, the process of identifying mul-tiple solutions using HAM is facilitated, even in the case forwhich the multiplicity in the numerical solution becomeselusive. Third, the solution of the nonlinear PDE is madepossible through the use of HAM, which could deal withboth nonlinear ODEs and nonlinear PDEs with equal level ofease. The main benefits of HAM stand, perhaps, in its abilityto solve a nonlinear equation using a parameter-free proce-dure that is so versatile that it can handle equally efficientlyeither a PDE or an ODE. Using similar tactics, our analyticalapproach can be further applied and incrementally modifiedto help in the identification of multiple steady and time-dependent solutions of a variety of nonlinear equations thatarise in fluid mechanics, especially in the field of global flowstability.

ACKNOWLEDGMENTS

H.X., Z.-L.L., and S.-J.L. extend their sincere apprecia-tion to the National Natural Science Foundation of China forthe support through Grant Nos. 10972036, 50739004, and10872129. S.-J.L. thanks the State Key Laboratory of OceanEngineering for the partial support through Grant No.GKZD010002. J.-Z.W. is thankful for the partial support re-ceived from the Sate Key Laboratory for Turbulence and

η

F(η

)/R

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

α = 5α = 0α =-5

FIG. 13. Spatial variation in the normal velocity profile F�� /R using a40th-order HAM approximation and three fixed values of � and R=5.

η

F(η

,t)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-6

-4

-2

0

2

4

6

t = 0.0, 0.5, 1.0, 2.0, 5.0

FIG. 14. Spatial variation in the normal velocity F� , t� using a �30,30�order homotopy-Padé approximation for some values of t and A=5. Here,��t� evolves exponentially from �1 to �2.

053601-15 Homotopy based solutions of the Navier–Stokes equations Phys. Fluids 22, 053601 �2010�

Complex Systems, Peking University, China. Finally, J.M. isdeeply grateful for the support received from the NationalScience Foundation, Grant No. CMMI-0928762, and throughthe travel supplement derived from the International Re-search and Education in Engineering �IREE� program onGrant No. CMMI-0353518.

APPENDIX: GENERAL PROCEDURE FOR SOLVINGTHE UNSTEADY EXPANSION RATIO PROBLEM

In what follows, we describe the mathematical stepsneeded to apply HAM to the porous channel flow problemwith a time-dependent expansion ratio ��t�. We thus considerthe unsteady PDE given by Eq. �21�. The attendant analysisencompasses the solution to Eq. �16�, which can be recov-ered for ��t�= 12 , ̇�t�=const. The description to follow canthus be applied to both steady and unsteady cases.

To begin, the form of the analytical solutions to Eq. �21�must be posited in term of suitable functions. Depending onthe character of �t� or ��t�, different solution expressionsare required. For the type I family of solutions that we seekhere, three cases arise depending on the behavior of ��t�. For��t�=const, F becomes independent of time; nonetheless, topresent the procedure systematically for the three cases athand, we still include t as we put

F�,t� =

i=0

+�

Bii. �A1�

Next, we consider the unsteady ��t�=�0 exp�−t�+�1�1−exp�−t��; the solution expression for the exponentially de-caying behavior may be readily expressed as

F�,t� =

i=0

�

j=0

�

k=0

�

Bi,jk itj exp�− kt� . �A2�

Finally, for ��t�=�1+ ��1−�0�t / �1+ t�, the series representa-tion for the algebraically varying expansion ratio may bewritten as

F�,t� =

i=0

�

j=0

�

k=0

�

s=0

�

Bi,jk,sitj

1

�1 + x�k�ln�1 + t��s. �A3�

Here Bi, Bi,jk , and Bi,j

k,s are undetermined coefficients that wemust seek to determine.

According to the homotopy-based approach, one needsto introduce an initial guess function of the form prescribedby the solution expressions while satisfying the problem’sboundary conditions. For all three cases, a suitable choicecorresponds to the leading order asymptotic solution ob-tained by Berman1 at small Reynolds numbers, specifically

�0�,t� =12R�3 −

2� . �A4�

Note that the above equation satisfies Eq. �11� and may beshown to be a viable candidate for the initial approximationof F� , t�. An auxiliary linear operator L is then defined tothe extent of capturing the highest order in Eq. �21�. This is

L���,t�� =�4��,t�

�4, �A5�

which has the property

L�C0 + C1 + C22 + C33� = 0, �A6�

where �C0 ,C1 ,C2 ,C3� are integral constants. This represen-tation satisfies the rule of solution expression through whichthe linear approximation is constructed from the set of basefunctions. At this point, a homotopy mapping relation maybe implemented of the type

F:,t � �0,1� � H���,t;0� = ��,t�;�A7�

��,t;1� = F�,t� .

This transformation leads to the HAM deformation equationthat has been well established as

�1 − q�L���,t;q� − �0�,t�� = q�N���,t;q�� , �A8�

where q� �0,1� is an embedding parameter and � is a non-zero auxiliary parameter known as the convergence-controlparameter.27 Note that N is a nonlinear operator obtained byrecasting Eq. �21� into N��� ;q��=0. We, therefore, have

N���;q�� =�4��;q�

�4+ ��;q�

�3��;q��3

+3

2

t�t�

�2��;q��2

−���;q�

�

�2��;q��2

+1

2

t�t�

�3��;q��3

− �t��3��;q�

�2 � t.

�A9�

The corresponding assortment of boundary conditions trans-lates into

��0,t;q� = 0, �2��,t;q��2

=0

= 0,

�A10�

��1,t;q� = R, ���,t;q��

=1

= 0.

At this point, it may be instructive to note that when

�q = 0: L���,t;0� − �0�,t�� = 0,q = 1: N���,t;1�� = 0; ∀ � � 0. �A11�

Hence as we track the evolution of q across the range �0,1�,the solution of the nonlinear equation will transition fromthe initial approximation �0�� to the target function thatwill exactly satisfy Eq. �21�. Expanding �� , t ;q� in theMaclaurin series form with respect to q, we have

��,t;q� = �0�,t,0� +

m=1

+�

�m�,t�qm, �A12�

where

053601-16 Xu et al. Phys. Fluids 22, 053601 �2010�

�m�,t� =1

m!

�m��,t;q�

�qm

q=0. �A13�

For brevity, we introduce a vector

�� m = ��0�,t�,�1�,t�,�2�,t�, . . . ,�m�,t�� . �A14�

The next step consists of differentiating the HAM deforma-tion equation �A8� m times with respect to q, dividing by m!,and then setting q=0. This enables us to obtain the mth-orderdeformation equation

L��m�,t� − m�m−1�,t�� = �Rm��m−1�,t�� , �A15�

which is subject to the boundary conditions

= 0: �m = 0, �m� = 0; = 1: �m = 0, �m� = 0.

�A16�

In Eq. �A15�, the right-hand-side function Rm is given by

Rm��m−1�,t�� =1

�m − 1�!

�m−1N���,t;q��

�qm−1

q=0

= �m−1�� +

i=0

m−1

��i�m−1−i� + �i��m−i−i� �

+1

2

t�t��3�m−1� + �m−1� � − �t��t m−1� ,

�A17�

where the prime denotes differentiation with respect to .By substituting the form of Eq. �A6�, the general solu-

tion for Eq. �A15� can be extracted. Using �m� � , t� to denote

a particular solution of Eq. �A15�, one collects

�m�,t� = m�m−1�,t� + �m� �,t� + C0

m + C1m + C2

m2

+ C3m3 �m � 0� , �A18�

where C0m, C1

m, C2m, and C3

m can be determined from theboundary conditions in Eq. �A16�. After some algebra, onegets

C0m = − �m

� �0,t�, C1m = − �m

� �1,t� − C0m − C2

m − C3m,

C2m = − 12�m

���0,t�, �A19�

C3m = 12 ��m

� �1,t� − �m���1,t� + C0

m + C2m� .

It can thus be seen that the linear relation �A15� may besolved recursively, one order at a time, for m=1,2 ,3 , . . .. Wenow recall that, according to the definition �A5� of L, thehigh-order deformation equation �A15� is linear. Then, fol-lowing the HAM procedure, a nonlinear PDE, such as Eq.�21�, may be transformed into an infinite number of linearODEs. The present approach guarantees the convergence ofthe series solution by providing the freedom to select a suit-able convergence-control parameter � or to use thehomotopy-Padé technique.28–31

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